# REDIRECT Sheaf ( mathematics )# The étale space of a sheaf

In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a co-chain complex.

From its beginning in topology, this idea became a dominant method in the mathematics of the second half of the twentieth century ; from the initial idea of homology as a topologically invariant relation on chains, the range of applications of homology and cohomology theories has spread out over geometry and abstract algebra.

In mathematics, de Rham cohomology ( after Georges de Rham ) is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapted to computation and the concrete representation of cohomology classes.

In mathematics, a characteristic class is a way of associating to each principal bundle on a topological space X a cohomology class of X.

In mathematics, particularly algebraic topology and homology theory, the Mayer – Vietoris sequence is an algebraic tool to help compute algebraic invariants of topological spaces, known as their homology and cohomology groups.

In mathematics, Hodge theory, named after W. V. D. Hodge, is one aspect of the study of differential forms of a smooth manifold M. More specifically, it works out the consequences for the cohomology groups of M, with real coefficients, of the partial differential equation theory of generalised Laplacian operators associated to a Riemannian metric on M.

In mathematics, the Koszul complex was first introduced to define a cohomology theory for Lie algebras, by Jean-Louis Koszul ( see Lie algebra cohomology ).

In mathematics, the Poincaré duality theorem named after Henri Poincaré, is a basic result on the structure of the homology and cohomology groups of manifolds.

In mathematics, the plus construction is a method for simplifying the fundamental group of a space without changing its homology and cohomology groups.

In mathematics, particularly in algebraic topology, Alexander – Spanier cohomology is a cohomology theory for topological spaces, introduced by for the special case of compact metric spaces, and by for all topological spaces, based on a suggestion of A. D. Wallace.

In mathematics, the Borel – Weil-Bott theorem is a basic result in the representation theory of Lie groups, showing how a family of representations can be obtained from holomorphic sections of certain complex vector bundles, and, more generally, from higher sheaf cohomology groups associated to such bundles.

Motivic cohomology is a cohomological theory in mathematics, the existence of which was first conjectured by Alexander Grothendieck during the 1960s.

In mathematics, more specifically in cohomology theory, a-cocycle in the cochain group is associated with a unique equivalence class known as the cocycle class or coclass of

In mathematics, specifically algebraic topology, Čech cohomology is a cohomology theory based on the intersection properties of open covers of a topological space.

* Brown – Peterson cohomology, a generalized cohomology theory in mathematics

In mathematics, the Chern – Weil homomorphism is a basic construction in the Chern – Weil theory, relating for a smooth manifold M the curvature of M to the de Rham cohomology groups of M, i. e., geometry to topology.

In stable homotopy theory, a branch of mathematics, Morava K-theory is one of a collection of cohomology theories introduced in algebraic topology by Jack Morava in unpublished preprints in the early 1970s.

In mathematics, complex cobordism is a generalized cohomology theory related to cobordism of manifolds.

Other examples are readily found in different areas of mathematics, for example, vector addition, matrix multiplication and conjugation in groups.

The Langlands program is a far-reaching web of these ideas of ' unifying conjectures ' that link different subfields of mathematics, e. g. number theory and representation theory of Lie groups ; some of these conjectures have since been proved.

In mathematics, the classification of the finite simple groups is a theorem stating that every finite simple group belongs to one of four categories described below.

The classification theorem has applications in many branches of mathematics, as questions about the structure of finite groups ( and their action on other mathematical objects ) can sometimes be reduced to questions about finite simple groups.

According to Husserl, this view of logic and mathematics accounted for the objectivity of a series of mathematical developments of his time, such as n-dimensional manifolds ( both Euclidean and non-Euclidean ), Hermann Grassmann's theory of extensions, William Rowan Hamilton's Hamiltonians, Sophus Lie's theory of transformation groups, and Cantor's set theory.

Finite fields have applications in many areas of mathematics and computer science, including coding theory, LFSRs, modular representation theory, and the groups of Lie type.

In mathematics, given two groups ( G, *) and ( H, ·), a group homomorphism from ( G, *) to ( H, ·) is a function h: G → H such that for all u and v in G it holds that

In areas of mathematics where one considers groups endowed with additional structure, a homomorphism sometimes means a map which respects not only the group structure ( as above ) but also the extra structure.

Fischbein and Schnarch ( 1997 ) administered a questionnaire to five groups: students in grades 5, 7, 9, 11, and college students specializing in teaching mathematics.

* Homology ( mathematics ), a procedure to associate a sequence of abelian groups or modules with a given mathematical object

In mathematics, a Lie algebra (, not ) is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds.

Lie groups represent the best-developed theory of continuous symmetry of mathematical objects and structures, which makes them indispensable tools for many parts of contemporary mathematics, as well as for modern theoretical physics.

Lie groups occur in abundance throughout mathematics and physics.

Although today Sophus Lie is rightfully recognized as the creator of the theory of continuous groups, a major stride in the development of their structure theory, which was to have a profound influence on subsequent development of mathematics, was made by Wilhelm Killing, who in 1888 published the first paper in a series entitled

The ubiquity of groups in numerous areas within and outside mathematics makes them a central organizing principle of contemporary mathematics.

In mathematics, model theory is the study of ( classes of ) mathematical structures ( e. g. groups, fields, graphs, universes of set theory ) using tools from mathematical logic.

His first study of mathematics, which would later lead to its incorporation into his art works, began with George Pólya's academic paper on plane symmetry groups sent to him by his brother Berend.

In mathematics, profinite groups are topological groups that are in a certain sense assembled from finite groups ; they share many properties with their finite quotients.

The symmetric group is important to diverse areas of mathematics such as Galois theory, invariant theory, the representation theory of Lie groups, and combinatorics.

In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.

The concept appears in most fields of mathematics, including geometry and mathematical analysis, much like the concept of groups does.

The study of topology in mathematics extends all over through point set topology, algebraic topology, differential topology, and all the related paraphernalia, such as homology theory, homotopy theory.

It is also commonly used in mathematics in algebraic solutions representing quantities such as angles.

In mathematics, an algebraic number is a number that is a root of a non-zero polynomial in one variable with rational coefficients ( or equivalently — by clearing denominators — with integer coefficients ).

Associative operations are abundant in mathematics ; in fact, many algebraic structures ( such as semigroups and categories ) explicitly require their binary operations to be associative.

In his youth he went to the continent and taught mathematics in Paris, where he published or edited, between the years 1612 and 1619, various geometric and algebraic tracts.

Initially a study of systems of polynomial equations in several variables, the subject of algebraic geometry starts where equation solving leaves off, and it becomes even more important to understand the intrinsic properties of the totality of solutions of a system of equations, than to find a specific solution ; this leads into some of the deepest areas in all of mathematics, both conceptually and in terms of technique.

In mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed.

Some branches of mathematics such as algebraic geometry, typically influenced by the French school of Bourbaki, use the term quasi-compact for the general notion, and reserve the term compact for topological spaces that are both Hausdorff and quasi-compact.

Even though equivalence relations are as ubiquitous in mathematics as order relations, the algebraic structure of equivalences is not as well known as that of orders.

In mathematics and abstract algebra, a group is the algebraic structure, where is a non-empty set and denotes a binary operation called the group operation.

In modern mathematics, the theory of fields ( or field theory ) plays an essential role in number theory and algebraic geometry.

In mathematics, more specifically algebraic topology, the fundamental group ( defined by Henri Poincaré in his article Analysis Situs, published in 1895 ) is a group associated to any given pointed topological space that provides a way of determining when two paths, starting and ending at a fixed base point, can be continuously deformed into each other.

( See areas of mathematics and algebraic geometry.

" Kronecker even objected to Cantor's proofs that the algebraic numbers are countable, and that the transcendental numbers are uncountable, results now included in a standard mathematics curriculum.

Many areas of mathematics and computer science have been brought to bear on the problem, including elliptic curves, algebraic number theory, and quantum computing.

Thus three major themes in 19th century mathematics were combined by Lie in creating his new theory: the idea of symmetry, as exemplified by Galois through the algebraic notion of a group ; geometric theory and the explicit solutions of differential equations of mechanics, worked out by Poisson and Jacobi ; and the new understanding of geometry that emerged in the works of Plücker, Möbius, Grassmann and others, and culminated in Riemann's revolutionary vision of the subject.

The development of algebraic geometry from its classical to modern forms is a particularly striking example of the way an area of mathematics can change radically in its viewpoint, without making what was correctly proved before in any way incorrect ; of course mathematical progress clarifies gaps in previous proofs, often by exposing hidden assumptions, which progress has revealed worth conceptualizing.

In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element.

In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single associative binary operation and an identity element.

For example, rather than just getting a passing grade for mathematics, a student might be assessed as level 4 for number sense, level 5 for algebraic concepts, level 3 for measurement skills, etc.

In advanced mathematics, polynomials are used to construct polynomial rings, a central concept in abstract algebra and algebraic geometry.

In mathematics, especially in abstract algebra, a quasigroup is an algebraic structure resembling a group in the sense that " division " is always possible.

* Ring ( mathematics ), an algebraic structure